N OU < V4 R0 2

ak

- D

pda 1

"

€ (OQ "^ = U4 0 LR =

In general, MAP estimator is computationally efficient and is

not sensitive to irregular sampling. However, it obeys the

Rayleigh resolution limit, i.e. it has almost no super-resolution

capability.

32.2 Nonlinear Least Squares (NLS)

Assuming the presence of K scatterers inside a pixel with

elevations of s=[s, 8a, ae the under-determined

system model (1) reduces to the following over-determined

problem:

$us =(R*(s)R(s)) R"(s)g 3)

For Gaussian white noise, NLS is identical to the maximum

likelihood estimator (MLE). It is therefore theoretically the best

estimator for our application if and only if the data closely

agree with the assumed model. Due to the large computational

effort to the multi-dimensional search, the NP-hard NLS is not

recommended for practical data processing.

3.2.3 SLIMMER

The "Scale-down by L1 norm Minimization, Model selection,

and Estimation Reconstruction" (SLIMMER, pronounced

"slimmer") algorithm is a spectral estimator firstly proposed in

(Zhu and Bamler, 2010b). It consists of three main steps: 1) a

dimensionality scale-down by L1 norm minimization, 2) model

selection and 3) linear parameter estimation. In case there is no

prior knowledge about the number of scatterers inside the pixel

and in the presence of measurement noise, the sparse

reconstruction of (1) is give by the following L1- L2 norm

minimization:

adr) @

The L;-L, norm minimization step shrinks R dramatically and

gives a first sparse estimate of y. Due to the fact the following

two effects (Zhu, 2011): 1) for our application, RIP and

incoherence are violated for several reasons 2) - The L1 norm

approximation of the NP-hard LO norm regularization

introduces amplitude bias, This estimate may still contain the

outliers. Therefore, a further model order selection and

parameter estimation step are followed to refine the sparse

estimates obtained from (5).

SLIMMER offers an aesthetic non-parametric realization of the

NP-hard NLS estimator. As an efficient estimator, it is

demonstrated to provide a super-resolution capability reaches

the fundamental bounds of all spectral estimators.

Younes = 918 min {lg -Ry

3.3 Tomographic SAR Reconstruction with Colored Noise

Based on the discussion in Section 2.2, i.e. the different data

quality possessed by repeat- and single-pass data can be

therefore characterized by the noise variance matrix C,, , the

aforementioned estimators are extended to the colored noise

case as following:

e MAP Estimator

f,» -(R" C2 R41) R*Czg (5)

* Nonlinear Least Squares (NLS)

faus-(R'(sCZR(s) R*()C2g ©

e SLIMMER

hu cargmin((g-Ry) C?(g-Rr)e2.|rh] — C

International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XXXIX-B7, 2012

XXII ISPRS Congress, 25 August — 01 September 2012, Melbourne, Australia

3.4 Cramér-Rao Lower Bound of the Elevation Estimates

Under colored noise, the Cramér-Rao Lower Bound (CRLB) of

the elevation estimates in presence of only a single scatterer is

given by:

Ar

S mr E (8)

4r 42 |V SNR, 6,

where SNR, stands for the signal-to-noise ratio (SNR) of the

n? acquisition (n 21,...,N ). 6, is the re-weighted standard

deviation of the baseline distribution:

Y.SNR,b; (Y,SNR,b

Ge | Az — (9)

YSNR, | ESNR,

Compared to the CRLB under Gaussian white noise 0 (i.e. all

data possesses the same SNR ), the N- SNR dependent factor is

replaced by D SNR, to account for variant data quality and the

standard deviation of the baseline distribution o, is

accordingly replaced by the &, depends on the sampling

position b, re-weighted according to the data quality SNR, . It

tells that, in case of adding several high quality TanDEM-X

data to an existing TerraSAR-X stack, the elevation estimation

accuracy improvement depends not only on the number of

TanDEM-X images, but also the corresponding baseline

distribution. Le. widely spread baselines are preferred.

Considering a mixed stack consisting of N, images with a

higher SNR of SNR, and N, images with a lower SNR of

SNR, , let’s assume the standard deviation of the baseline

distribution for each sub-stack are o,, and o,, , eq. (8) can be

approximated by:

o EE (10)

YW ve,

where,

i=12 (11)

Ar

GET)

"4n A2 JN, SNR, o,,

4. EXPERIEMENNTAL RESULTS

A reasonable data quality assumption for a mixed repeat- and

single-pass data stack is that we have a mixed stack consisting

of N, images with a higher SNR of SNR, and N, images

with a lower SNR of SNR, .

The data is simulated using the regularly distributed elevation

aperture shown in Fig.5 (25 images, elevation resolution

p, = 40.5m ) with the following two cases:

e TSX: multi-pass data stack case, i.e. 25 images with

SNR=5dB;

e TDX: mixed stack, 20 images with SNR of 5dB

(elevation aperture positions marked as black) and 5

images with SNR=20dB (marked as green).

Fig.6 shows comparison of the reflectivity profiles along

elevation direction reconstructed by MAP (blue) and

SLIMMER (red) using the data stack of TSX and TDX cases

(right). The true elevations of the two scatterers are 10 and 25

meter, respectively, an i.e. elevation distance is 0.4 times of the

Rayleigh resolution limit. It is obvious to see the better sidelobe

suppression for the non-parametric estimator and better super-